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 A005565 Number of walks on square lattice. (Formerly M5087) 3
 20, 75, 189, 392, 720, 1215, 1925, 2904, 4212, 5915, 8085, 10800, 14144, 18207, 23085, 28880, 35700, 43659, 52877, 63480, 75600, 89375, 104949, 122472, 142100, 163995, 188325, 215264, 244992, 277695, 313565, 352800, 395604, 442187, 492765, 547560, 606800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 R. K. Guy, Letter to N. J. A. Sloane, May 1990 R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = 1/4*(n^4+14n^3+69n^2+136n+80). G.f.: (20-25x+14x^2-3x^3)/(1-x)^5. - Ralf Stephan, Apr 23 2004 a(n) = binomial(n+4,2)^2 - binomial(n+4,1)^2. - Gary Detlefs, Nov 22 2011 Using two consecutive triangular numbers t(n) and t(n+1), starting at n=3, compute the determinant of a 2 X 2 matrix with the first row t(n), t(n+1) and the second row t(n+1), 2*t(n+1). This gives (n+1)^2*(n-2)*(n+2)/4 = a(n-3). - J. M. Bergot, May 17 2012 MAPLE seq(add (k^3-n^2, k =0..n), n=4..28 ); # Zerinvary Lajos, Aug 26 2007 A005565:=(-20+25*z-14*z**2+3*z**3)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA CoefficientList[Series[(20-25x+14x^2-3x^3)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 24 2012 *) LinearRecurrence[{5, -10, 10, -5, 1}, {20, 75, 189, 392, 720}, 40] (* Harvey P. Dale, Dec 04 2020 *) PROG (PARI) a(n)=(n^4+14*n^3+69*n^2+136*n)/4+20 \\ Charles R Greathouse IV, Nov 22 2011 (Magma) [1/4*(n^4+14*n^3+69*n^2+136*n+80): n in [0..40]]; // Vincenzo Librandi, May 24 2012 CROSSREFS Sequence in context: A237617 A000529 A238027 * A320484 A066126 A228025 Adjacent sequences: A005562 A005563 A005564 * A005566 A005567 A005568 KEYWORD nonn,walk,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified September 25 20:01 EDT 2023. Contains 365649 sequences. (Running on oeis4.)